Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [322,4,Mod(1,322)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("322.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 322 = 2 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 322.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(18.9986150218\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 126x^{4} + 262x^{3} + 3854x^{2} - 12552x + 9792 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - x^{5} - 126x^{4} + 262x^{3} + 3854x^{2} - 12552x + 9792 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 64\nu + 78 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 3\nu^{2} + 67\nu - 207 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 5\nu^{4} + 130\nu^{3} - 434\nu^{2} - 3906\nu + 7452 ) / 36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - \nu^{4} + 124\nu^{3} - 32\nu^{2} - 3900\nu + 5904 ) / 18 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - \beta _1 + 43 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} + 64\beta _1 - 78 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{5} + 6\beta_{4} + 67\beta_{3} + 64\beta_{2} - 130\beta _1 + 2701 \)
|
| \(\nu^{5}\) | \(=\) |
\( -15\beta_{5} - 6\beta_{4} - 99\beta_{3} + 276\beta_{2} + 4198\beta _1 - 7845 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | −6.71266 | 4.00000 | 19.3030 | −13.4253 | 7.00000 | 8.00000 | 18.0598 | 38.6060 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −6.09966 | 4.00000 | −11.5685 | −12.1993 | 7.00000 | 8.00000 | 10.2058 | −23.1371 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 3.40643 | 4.00000 | 3.67006 | 6.81286 | 7.00000 | 8.00000 | −15.3962 | 7.34011 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 3.87764 | 4.00000 | 11.9721 | 7.75528 | 7.00000 | 8.00000 | −11.9639 | 23.9442 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 8.79154 | 4.00000 | 9.67477 | 17.5831 | 7.00000 | 8.00000 | 50.2911 | 19.3495 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 9.73671 | 4.00000 | −21.0514 | 19.4734 | 7.00000 | 8.00000 | 67.8035 | −42.1028 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(23\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 322.4.a.i | ✓ | 6 |
| 7.b | odd | 2 | 1 | 2254.4.a.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.4.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2254.4.a.m | 6 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 13T_{3}^{5} - 56T_{3}^{4} + 1070T_{3}^{3} - 422T_{3}^{2} - 21064T_{3} + 46296 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(322))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( T^{6} - 13 T^{5} + \cdots + 46296 \)
|
| $5$ |
\( T^{6} - 12 T^{5} + \cdots + 1998336 \)
|
| $7$ |
\( (T - 7)^{6} \)
|
| $11$ |
\( T^{6} - 58 T^{5} + \cdots - 8433408 \)
|
| $13$ |
\( T^{6} + \cdots - 7073110016 \)
|
| $17$ |
\( T^{6} + \cdots - 32215087104 \)
|
| $19$ |
\( T^{6} + \cdots - 26652016640 \)
|
| $23$ |
\( (T + 23)^{6} \)
|
| $29$ |
\( T^{6} + \cdots - 7731205376112 \)
|
| $31$ |
\( T^{6} + \cdots - 3085301866896 \)
|
| $37$ |
\( T^{6} + \cdots + 254726498944 \)
|
| $41$ |
\( T^{6} + \cdots - 12264071508576 \)
|
| $43$ |
\( T^{6} + \cdots - 244907309621248 \)
|
| $47$ |
\( T^{6} + \cdots - 536861089431216 \)
|
| $53$ |
\( T^{6} + \cdots + 3868373919744 \)
|
| $59$ |
\( T^{6} + \cdots - 12\!\cdots\!28 \)
|
| $61$ |
\( T^{6} + \cdots + 62\!\cdots\!28 \)
|
| $67$ |
\( T^{6} + \cdots - 452876890817536 \)
|
| $71$ |
\( T^{6} + \cdots - 50\!\cdots\!52 \)
|
| $73$ |
\( T^{6} + \cdots + 6143805807072 \)
|
| $79$ |
\( T^{6} + \cdots + 11\!\cdots\!40 \)
|
| $83$ |
\( T^{6} + \cdots - 13\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots - 40\!\cdots\!52 \)
|
| $97$ |
\( T^{6} + \cdots + 10\!\cdots\!12 \)
|
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